Most reboots don't have any major cultural impact because, quite frankly, they don't have as massive the audience as event movies that are highly anticipated. If your going to compare Webb to Nolan then you have to use Batman Begins which was Nolan's introduction reboot film like ASM was and you'll notice BB didn't really stay in the cultural memory like TDK or SM1 either.

But both those movies had some of the biggest arch enemies with built up anticipation. Of course they'll make more cultural impact than The Lizard or Ras Al Ghul in a reboot film.

It does stay around as the first reboot and also more now since Nolan's trilogy is complete. TAS-M will be lucky to have the latter.